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Theoretical and Natural Science

Degeneracy and Symmetry of the Infinite Potential Well

Yuanbei Sun

Published Oct 23, 2025• 0 citations

Abstract

This paper solves the wavefunctions and energy level formulas of two types of three-dimensional infinite deep cubic potential wells and three-dimensional infinite deep spher-ical potential wells, and studies their degeneracy to elucidate how symmetry governs energy level degeneracy in quantum systems. For the cubic potential well, the potential energy inside the cube is zero, and outside the cube is infinite. Solving the Schrödinger equation shows that the energy is determined by the total of the squares of three quantum numbers (nx, ny, nz). Degeneracy occurs when different combinations of quantum numbers produce the same sum of squares, which reflects cubic symmetry, i.e., the equivalence of the x, y, and z directions, such that permuted states are physically identical. For the spherical potential well, the poten-tial energy inside the sphere is zero, and outside the sphere is infinite. The wavefunction can be devided into angular and radial parts. The energy depends on the radial quantum number n and the angular momentum quantum number l, and degeneracy arises from the magnetic quantum number m, which ranges from −l to l. Since the potential energy of the spherical potential well is independent of direction, it has spherical symmetry, which leads to degen-eracy of magnitude 2l + 1-fold. To summarize, the degeneracy of the cubic potential well originates from its cubic symmetry, while that of the spherical potential well originates from its spherical symmetry. This highlights the central role of symmetry in quantum energy level degeneracy.

DOI: 10.54254/2753-8818/2026.hz28374

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By solving the Schrodinger equations in three-dimensional (3D) and two-dimensional (2D) finite depth center spherically symmetric square potential well, the eigenequations of bounded energy levels and their distributions are obtained, and the analytical solution for any bound state wave function is given. Both the eigenequations of the energy levels as well as the stationary state wave functions are closely related to the (spherical) Bessel functions. It is found that for the 3D case, there exists a bound solution when 2μa2V0/ħ2>π2/4 is present. While for the 2D case, there exists a bound state if 2μa2V0/ħ2>0.1993. Finally, numerical methods are used to analyze the characteristics of probability density distribution of different quantum states. It is found that in both three-dimensional and two-dimensional situations, the maximum number of probability density distributions inside the well corresponds exactly to the quantum number n. The larger n or the smaller α is, the higher the probability of discovering particles outside the well. In addition, the angular momentum quantum number l or m can characterize the degree to which particles deviate from the center or the uniform distribution.

Although the general properties of the phase shift of the three-dimensional spherically symmetrical square well potential are described in most textbooks on quantum mechanics, specific graphical representations of this function are rare. In this paper we present a number of phase shifts, as a function of energy, for the various values of the angular momentum quantum number. We discuss the general features of these curves. In particular, we consider what happens to the phase shifts if one varies the depth of the well, not only for near-zero energy, but for the entire range of energies. We describe how the bound levels of a deep well are related to the virtual levels of the shallow well. We find points of triple coincidence for curves belonging to three successive l values for given well depth. A proof of this phenomenon is given.

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The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).

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